The kinetic equation for soliton gases, and the (Euler-scale) GHD equation for many-body integrable models, are strikingly similar, something already pointed out both in the contexts of GHD and dispersive hydrodynamics, see e.g. [1]}, [2]}, [3]}. Much like for soliton gases, in GHD it can be seen as an effective propagation equation for the stable quasi-particles of many-body integrable systems, as advocated in [4]}. Semi-classical arguments made in [1]} further argue that the GHD equation can be recovered using Zakharov's 1971 idea that scattering shifts lead to modified velocities. This idea is also at the root of the early derivation in the 1970's of the hydrodynamic equations for the hard rod gas [6]}, which GHD generalises. But besides these remarks, the precise mathematical relation between the kinetic theory of soliton gases in dispersive hydrodynamics, and GHD, has not been established so far.
